Properties

Label 531.8.a.b.1.6
Level $531$
Weight $8$
Character 531.1
Self dual yes
Analytic conductor $165.876$
Analytic rank $1$
Dimension $16$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 531.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(165.876448532\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 6 x^{15} - 1493 x^{14} + 8791 x^{13} + 890490 x^{12} - 5107725 x^{11} - 269092298 x^{10} + 1488374176 x^{9} + 42885295136 x^{8} - 226132003872 x^{7} - 3353576629440 x^{6} + 16796366777600 x^{5} + 99470801612800 x^{4} - 494039551757568 x^{3} - 493048066650624 x^{2} + 3193975642099712 x - 2385018853548032\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 177)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-7.02227\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q-7.02227 q^{2} -78.6878 q^{4} +266.773 q^{5} +665.758 q^{7} +1451.42 q^{8} +O(q^{10})\) \(q-7.02227 q^{2} -78.6878 q^{4} +266.773 q^{5} +665.758 q^{7} +1451.42 q^{8} -1873.35 q^{10} -2997.05 q^{11} +11110.8 q^{13} -4675.13 q^{14} -120.197 q^{16} -17329.7 q^{17} -7569.90 q^{19} -20991.8 q^{20} +21046.1 q^{22} +84837.6 q^{23} -6956.93 q^{25} -78023.0 q^{26} -52387.0 q^{28} -59721.5 q^{29} -56202.0 q^{31} -184937. q^{32} +121694. q^{34} +177607. q^{35} -375672. q^{37} +53157.8 q^{38} +387199. q^{40} -560321. q^{41} -599895. q^{43} +235831. q^{44} -595752. q^{46} +508735. q^{47} -380309. q^{49} +48853.4 q^{50} -874284. q^{52} -95112.1 q^{53} -799533. q^{55} +966292. q^{56} +419380. q^{58} -205379. q^{59} +1.08522e6 q^{61} +394665. q^{62} +1.31406e6 q^{64} +2.96407e6 q^{65} -731050. q^{67} +1.36364e6 q^{68} -1.24720e6 q^{70} -4.30967e6 q^{71} -804495. q^{73} +2.63807e6 q^{74} +595658. q^{76} -1.99531e6 q^{77} -3.53260e6 q^{79} -32065.4 q^{80} +3.93472e6 q^{82} +8.87854e6 q^{83} -4.62312e6 q^{85} +4.21262e6 q^{86} -4.34997e6 q^{88} +5.83811e6 q^{89} +7.39710e6 q^{91} -6.67568e6 q^{92} -3.57247e6 q^{94} -2.01945e6 q^{95} +1.47129e7 q^{97} +2.67063e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 6q^{2} + 974q^{4} + 68q^{5} - 2343q^{7} - 819q^{8} + O(q^{10}) \) \( 16q + 6q^{2} + 974q^{4} + 68q^{5} - 2343q^{7} - 819q^{8} - 3479q^{10} - 898q^{11} - 8172q^{13} + 13315q^{14} + 3138q^{16} + 44985q^{17} - 40137q^{19} - 130657q^{20} + 109394q^{22} + 2833q^{23} + 285746q^{25} + 129420q^{26} + 112890q^{28} - 144375q^{29} - 141759q^{31} + 36224q^{32} - 341332q^{34} + 78859q^{35} - 297971q^{37} - 329075q^{38} - 203048q^{40} - 659077q^{41} - 1431608q^{43} - 254916q^{44} + 873113q^{46} + 1574073q^{47} + 1893545q^{49} - 302533q^{50} - 4972548q^{52} - 587736q^{53} - 4624036q^{55} + 5798506q^{56} - 6991380q^{58} - 3286064q^{59} - 6117131q^{61} + 11570258q^{62} - 19063011q^{64} + 5335514q^{65} - 16518710q^{67} + 17284669q^{68} - 39189486q^{70} + 10882582q^{71} - 21097441q^{73} + 16717030q^{74} - 40864952q^{76} + 3404601q^{77} - 3784458q^{79} + 27466195q^{80} - 24990117q^{82} + 1951425q^{83} - 23238675q^{85} + 35910572q^{86} - 27843055q^{88} - 10499443q^{89} + 699217q^{91} + 20062766q^{92} - 59358988q^{94} + 29236333q^{95} - 25158976q^{97} - 2120460q^{98} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −7.02227 −0.620686 −0.310343 0.950625i \(-0.600444\pi\)
−0.310343 + 0.950625i \(0.600444\pi\)
\(3\) 0 0
\(4\) −78.6878 −0.614748
\(5\) 266.773 0.954438 0.477219 0.878784i \(-0.341645\pi\)
0.477219 + 0.878784i \(0.341645\pi\)
\(6\) 0 0
\(7\) 665.758 0.733623 0.366812 0.930295i \(-0.380449\pi\)
0.366812 + 0.930295i \(0.380449\pi\)
\(8\) 1451.42 1.00225
\(9\) 0 0
\(10\) −1873.35 −0.592407
\(11\) −2997.05 −0.678922 −0.339461 0.940620i \(-0.610245\pi\)
−0.339461 + 0.940620i \(0.610245\pi\)
\(12\) 0 0
\(13\) 11110.8 1.40263 0.701316 0.712851i \(-0.252596\pi\)
0.701316 + 0.712851i \(0.252596\pi\)
\(14\) −4675.13 −0.455350
\(15\) 0 0
\(16\) −120.197 −0.00733625
\(17\) −17329.7 −0.855502 −0.427751 0.903897i \(-0.640694\pi\)
−0.427751 + 0.903897i \(0.640694\pi\)
\(18\) 0 0
\(19\) −7569.90 −0.253193 −0.126597 0.991954i \(-0.540405\pi\)
−0.126597 + 0.991954i \(0.540405\pi\)
\(20\) −20991.8 −0.586739
\(21\) 0 0
\(22\) 21046.1 0.421397
\(23\) 84837.6 1.45392 0.726961 0.686679i \(-0.240932\pi\)
0.726961 + 0.686679i \(0.240932\pi\)
\(24\) 0 0
\(25\) −6956.93 −0.0890487
\(26\) −78023.0 −0.870594
\(27\) 0 0
\(28\) −52387.0 −0.450994
\(29\) −59721.5 −0.454713 −0.227357 0.973812i \(-0.573008\pi\)
−0.227357 + 0.973812i \(0.573008\pi\)
\(30\) 0 0
\(31\) −56202.0 −0.338833 −0.169416 0.985545i \(-0.554188\pi\)
−0.169416 + 0.985545i \(0.554188\pi\)
\(32\) −184937. −0.997699
\(33\) 0 0
\(34\) 121694. 0.530998
\(35\) 177607. 0.700198
\(36\) 0 0
\(37\) −375672. −1.21928 −0.609639 0.792679i \(-0.708686\pi\)
−0.609639 + 0.792679i \(0.708686\pi\)
\(38\) 53157.8 0.157154
\(39\) 0 0
\(40\) 387199. 0.956587
\(41\) −560321. −1.26968 −0.634839 0.772645i \(-0.718934\pi\)
−0.634839 + 0.772645i \(0.718934\pi\)
\(42\) 0 0
\(43\) −599895. −1.15063 −0.575315 0.817932i \(-0.695120\pi\)
−0.575315 + 0.817932i \(0.695120\pi\)
\(44\) 235831. 0.417366
\(45\) 0 0
\(46\) −595752. −0.902429
\(47\) 508735. 0.714741 0.357371 0.933963i \(-0.383673\pi\)
0.357371 + 0.933963i \(0.383673\pi\)
\(48\) 0 0
\(49\) −380309. −0.461797
\(50\) 48853.4 0.0552713
\(51\) 0 0
\(52\) −874284. −0.862265
\(53\) −95112.1 −0.0877547 −0.0438773 0.999037i \(-0.513971\pi\)
−0.0438773 + 0.999037i \(0.513971\pi\)
\(54\) 0 0
\(55\) −799533. −0.647988
\(56\) 966292. 0.735276
\(57\) 0 0
\(58\) 419380. 0.282234
\(59\) −205379. −0.130189
\(60\) 0 0
\(61\) 1.08522e6 0.612160 0.306080 0.952006i \(-0.400982\pi\)
0.306080 + 0.952006i \(0.400982\pi\)
\(62\) 394665. 0.210309
\(63\) 0 0
\(64\) 1.31406e6 0.626594
\(65\) 2.96407e6 1.33872
\(66\) 0 0
\(67\) −731050. −0.296951 −0.148476 0.988916i \(-0.547437\pi\)
−0.148476 + 0.988916i \(0.547437\pi\)
\(68\) 1.36364e6 0.525918
\(69\) 0 0
\(70\) −1.24720e6 −0.434603
\(71\) −4.30967e6 −1.42902 −0.714512 0.699623i \(-0.753351\pi\)
−0.714512 + 0.699623i \(0.753351\pi\)
\(72\) 0 0
\(73\) −804495. −0.242043 −0.121022 0.992650i \(-0.538617\pi\)
−0.121022 + 0.992650i \(0.538617\pi\)
\(74\) 2.63807e6 0.756790
\(75\) 0 0
\(76\) 595658. 0.155650
\(77\) −1.99531e6 −0.498073
\(78\) 0 0
\(79\) −3.53260e6 −0.806121 −0.403060 0.915173i \(-0.632054\pi\)
−0.403060 + 0.915173i \(0.632054\pi\)
\(80\) −32065.4 −0.00700199
\(81\) 0 0
\(82\) 3.93472e6 0.788072
\(83\) 8.87854e6 1.70439 0.852193 0.523227i \(-0.175272\pi\)
0.852193 + 0.523227i \(0.175272\pi\)
\(84\) 0 0
\(85\) −4.62312e6 −0.816523
\(86\) 4.21262e6 0.714180
\(87\) 0 0
\(88\) −4.34997e6 −0.680451
\(89\) 5.83811e6 0.877823 0.438912 0.898530i \(-0.355364\pi\)
0.438912 + 0.898530i \(0.355364\pi\)
\(90\) 0 0
\(91\) 7.39710e6 1.02900
\(92\) −6.67568e6 −0.893796
\(93\) 0 0
\(94\) −3.57247e6 −0.443630
\(95\) −2.01945e6 −0.241657
\(96\) 0 0
\(97\) 1.47129e7 1.63681 0.818405 0.574642i \(-0.194859\pi\)
0.818405 + 0.574642i \(0.194859\pi\)
\(98\) 2.67063e6 0.286631
\(99\) 0 0
\(100\) 547426. 0.0547426
\(101\) 4.85985e6 0.469352 0.234676 0.972074i \(-0.424597\pi\)
0.234676 + 0.972074i \(0.424597\pi\)
\(102\) 0 0
\(103\) −4.63498e6 −0.417944 −0.208972 0.977922i \(-0.567012\pi\)
−0.208972 + 0.977922i \(0.567012\pi\)
\(104\) 1.61264e7 1.40579
\(105\) 0 0
\(106\) 667902. 0.0544681
\(107\) 1.55941e7 1.23060 0.615301 0.788292i \(-0.289035\pi\)
0.615301 + 0.788292i \(0.289035\pi\)
\(108\) 0 0
\(109\) −554937. −0.0410442 −0.0205221 0.999789i \(-0.506533\pi\)
−0.0205221 + 0.999789i \(0.506533\pi\)
\(110\) 5.61454e6 0.402198
\(111\) 0 0
\(112\) −80022.2 −0.00538204
\(113\) 812959. 0.0530022 0.0265011 0.999649i \(-0.491563\pi\)
0.0265011 + 0.999649i \(0.491563\pi\)
\(114\) 0 0
\(115\) 2.26324e7 1.38768
\(116\) 4.69935e6 0.279534
\(117\) 0 0
\(118\) 1.44223e6 0.0808065
\(119\) −1.15374e7 −0.627616
\(120\) 0 0
\(121\) −1.05049e7 −0.539066
\(122\) −7.62074e6 −0.379960
\(123\) 0 0
\(124\) 4.42241e6 0.208297
\(125\) −2.26976e7 −1.03943
\(126\) 0 0
\(127\) −3.99070e7 −1.72877 −0.864383 0.502834i \(-0.832291\pi\)
−0.864383 + 0.502834i \(0.832291\pi\)
\(128\) 1.44443e7 0.608780
\(129\) 0 0
\(130\) −2.08145e7 −0.830928
\(131\) −4.33560e7 −1.68500 −0.842500 0.538696i \(-0.818917\pi\)
−0.842500 + 0.538696i \(0.818917\pi\)
\(132\) 0 0
\(133\) −5.03972e6 −0.185749
\(134\) 5.13363e6 0.184314
\(135\) 0 0
\(136\) −2.51527e7 −0.857429
\(137\) 3.31553e7 1.10162 0.550809 0.834632i \(-0.314319\pi\)
0.550809 + 0.834632i \(0.314319\pi\)
\(138\) 0 0
\(139\) −4.77383e7 −1.50770 −0.753850 0.657046i \(-0.771806\pi\)
−0.753850 + 0.657046i \(0.771806\pi\)
\(140\) −1.39755e7 −0.430445
\(141\) 0 0
\(142\) 3.02636e7 0.886976
\(143\) −3.32996e7 −0.952277
\(144\) 0 0
\(145\) −1.59321e7 −0.433996
\(146\) 5.64938e6 0.150233
\(147\) 0 0
\(148\) 2.95608e7 0.749549
\(149\) 7.91243e7 1.95956 0.979778 0.200086i \(-0.0641220\pi\)
0.979778 + 0.200086i \(0.0641220\pi\)
\(150\) 0 0
\(151\) −5.25091e7 −1.24112 −0.620562 0.784157i \(-0.713096\pi\)
−0.620562 + 0.784157i \(0.713096\pi\)
\(152\) −1.09871e7 −0.253764
\(153\) 0 0
\(154\) 1.40116e7 0.309147
\(155\) −1.49932e7 −0.323395
\(156\) 0 0
\(157\) −5.32097e7 −1.09734 −0.548671 0.836038i \(-0.684866\pi\)
−0.548671 + 0.836038i \(0.684866\pi\)
\(158\) 2.48069e7 0.500348
\(159\) 0 0
\(160\) −4.93364e7 −0.952241
\(161\) 5.64813e7 1.06663
\(162\) 0 0
\(163\) 6.98231e7 1.26282 0.631412 0.775448i \(-0.282476\pi\)
0.631412 + 0.775448i \(0.282476\pi\)
\(164\) 4.40904e7 0.780532
\(165\) 0 0
\(166\) −6.23475e7 −1.05789
\(167\) −8.36408e7 −1.38967 −0.694833 0.719171i \(-0.744522\pi\)
−0.694833 + 0.719171i \(0.744522\pi\)
\(168\) 0 0
\(169\) 6.07013e7 0.967375
\(170\) 3.24647e7 0.506805
\(171\) 0 0
\(172\) 4.72044e7 0.707347
\(173\) 2.42624e7 0.356265 0.178132 0.984007i \(-0.442994\pi\)
0.178132 + 0.984007i \(0.442994\pi\)
\(174\) 0 0
\(175\) −4.63163e6 −0.0653282
\(176\) 360237. 0.00498074
\(177\) 0 0
\(178\) −4.09967e7 −0.544853
\(179\) 1.12557e7 0.146686 0.0733430 0.997307i \(-0.476633\pi\)
0.0733430 + 0.997307i \(0.476633\pi\)
\(180\) 0 0
\(181\) 1.11658e7 0.139963 0.0699816 0.997548i \(-0.477706\pi\)
0.0699816 + 0.997548i \(0.477706\pi\)
\(182\) −5.19444e7 −0.638688
\(183\) 0 0
\(184\) 1.23135e8 1.45720
\(185\) −1.00219e8 −1.16373
\(186\) 0 0
\(187\) 5.19381e7 0.580819
\(188\) −4.00312e7 −0.439386
\(189\) 0 0
\(190\) 1.41811e7 0.149993
\(191\) 3.59428e7 0.373246 0.186623 0.982432i \(-0.440246\pi\)
0.186623 + 0.982432i \(0.440246\pi\)
\(192\) 0 0
\(193\) 2.78406e7 0.278758 0.139379 0.990239i \(-0.455489\pi\)
0.139379 + 0.990239i \(0.455489\pi\)
\(194\) −1.03318e8 −1.01595
\(195\) 0 0
\(196\) 2.99257e7 0.283889
\(197\) −8.97822e7 −0.836679 −0.418339 0.908291i \(-0.637388\pi\)
−0.418339 + 0.908291i \(0.637388\pi\)
\(198\) 0 0
\(199\) 1.97256e7 0.177437 0.0887183 0.996057i \(-0.471723\pi\)
0.0887183 + 0.996057i \(0.471723\pi\)
\(200\) −1.00974e7 −0.0892493
\(201\) 0 0
\(202\) −3.41272e7 −0.291320
\(203\) −3.97601e7 −0.333588
\(204\) 0 0
\(205\) −1.49479e8 −1.21183
\(206\) 3.25481e7 0.259412
\(207\) 0 0
\(208\) −1.33549e6 −0.0102901
\(209\) 2.26874e7 0.171898
\(210\) 0 0
\(211\) 1.59619e8 1.16976 0.584881 0.811119i \(-0.301141\pi\)
0.584881 + 0.811119i \(0.301141\pi\)
\(212\) 7.48416e6 0.0539470
\(213\) 0 0
\(214\) −1.09506e8 −0.763818
\(215\) −1.60036e8 −1.09820
\(216\) 0 0
\(217\) −3.74169e7 −0.248576
\(218\) 3.89692e6 0.0254756
\(219\) 0 0
\(220\) 6.29135e7 0.398350
\(221\) −1.92547e8 −1.19995
\(222\) 0 0
\(223\) −1.22127e8 −0.737471 −0.368736 0.929534i \(-0.620209\pi\)
−0.368736 + 0.929534i \(0.620209\pi\)
\(224\) −1.23123e8 −0.731935
\(225\) 0 0
\(226\) −5.70881e6 −0.0328977
\(227\) 1.05761e7 0.0600118 0.0300059 0.999550i \(-0.490447\pi\)
0.0300059 + 0.999550i \(0.490447\pi\)
\(228\) 0 0
\(229\) 5.44892e7 0.299838 0.149919 0.988698i \(-0.452099\pi\)
0.149919 + 0.988698i \(0.452099\pi\)
\(230\) −1.58931e8 −0.861313
\(231\) 0 0
\(232\) −8.66808e7 −0.455738
\(233\) −1.03393e8 −0.535484 −0.267742 0.963491i \(-0.586278\pi\)
−0.267742 + 0.963491i \(0.586278\pi\)
\(234\) 0 0
\(235\) 1.35717e8 0.682176
\(236\) 1.61608e7 0.0800334
\(237\) 0 0
\(238\) 8.10188e7 0.389553
\(239\) 9.19771e7 0.435800 0.217900 0.975971i \(-0.430079\pi\)
0.217900 + 0.975971i \(0.430079\pi\)
\(240\) 0 0
\(241\) −3.22890e8 −1.48592 −0.742958 0.669338i \(-0.766578\pi\)
−0.742958 + 0.669338i \(0.766578\pi\)
\(242\) 7.37679e7 0.334591
\(243\) 0 0
\(244\) −8.53939e7 −0.376325
\(245\) −1.01456e8 −0.440756
\(246\) 0 0
\(247\) −8.41076e7 −0.355137
\(248\) −8.15725e7 −0.339596
\(249\) 0 0
\(250\) 1.59389e8 0.645160
\(251\) −2.78313e8 −1.11090 −0.555451 0.831549i \(-0.687454\pi\)
−0.555451 + 0.831549i \(0.687454\pi\)
\(252\) 0 0
\(253\) −2.54263e8 −0.987099
\(254\) 2.80238e8 1.07302
\(255\) 0 0
\(256\) −2.69632e8 −1.00446
\(257\) 1.56960e8 0.576795 0.288398 0.957511i \(-0.406877\pi\)
0.288398 + 0.957511i \(0.406877\pi\)
\(258\) 0 0
\(259\) −2.50107e8 −0.894491
\(260\) −2.33236e8 −0.822978
\(261\) 0 0
\(262\) 3.04458e8 1.04586
\(263\) −2.42199e8 −0.820971 −0.410486 0.911867i \(-0.634641\pi\)
−0.410486 + 0.911867i \(0.634641\pi\)
\(264\) 0 0
\(265\) −2.53734e7 −0.0837563
\(266\) 3.53902e7 0.115292
\(267\) 0 0
\(268\) 5.75247e7 0.182550
\(269\) −5.70264e7 −0.178625 −0.0893126 0.996004i \(-0.528467\pi\)
−0.0893126 + 0.996004i \(0.528467\pi\)
\(270\) 0 0
\(271\) 2.57235e8 0.785124 0.392562 0.919726i \(-0.371589\pi\)
0.392562 + 0.919726i \(0.371589\pi\)
\(272\) 2.08298e6 0.00627617
\(273\) 0 0
\(274\) −2.32825e8 −0.683759
\(275\) 2.08503e7 0.0604571
\(276\) 0 0
\(277\) −6.22571e8 −1.75999 −0.879994 0.474984i \(-0.842454\pi\)
−0.879994 + 0.474984i \(0.842454\pi\)
\(278\) 3.35231e8 0.935809
\(279\) 0 0
\(280\) 2.57781e8 0.701775
\(281\) −3.00467e8 −0.807838 −0.403919 0.914795i \(-0.632352\pi\)
−0.403919 + 0.914795i \(0.632352\pi\)
\(282\) 0 0
\(283\) −4.37742e8 −1.14806 −0.574032 0.818833i \(-0.694621\pi\)
−0.574032 + 0.818833i \(0.694621\pi\)
\(284\) 3.39118e8 0.878490
\(285\) 0 0
\(286\) 2.33839e8 0.591065
\(287\) −3.73038e8 −0.931465
\(288\) 0 0
\(289\) −1.10019e8 −0.268117
\(290\) 1.11880e8 0.269375
\(291\) 0 0
\(292\) 6.33039e7 0.148796
\(293\) 3.94385e8 0.915976 0.457988 0.888958i \(-0.348570\pi\)
0.457988 + 0.888958i \(0.348570\pi\)
\(294\) 0 0
\(295\) −5.47897e7 −0.124257
\(296\) −5.45257e8 −1.22203
\(297\) 0 0
\(298\) −5.55632e8 −1.21627
\(299\) 9.42614e8 2.03932
\(300\) 0 0
\(301\) −3.99385e8 −0.844129
\(302\) 3.68733e8 0.770349
\(303\) 0 0
\(304\) 909879. 0.00185749
\(305\) 2.89509e8 0.584269
\(306\) 0 0
\(307\) −2.75368e8 −0.543162 −0.271581 0.962416i \(-0.587546\pi\)
−0.271581 + 0.962416i \(0.587546\pi\)
\(308\) 1.57006e8 0.306189
\(309\) 0 0
\(310\) 1.05286e8 0.200727
\(311\) 2.81733e8 0.531100 0.265550 0.964097i \(-0.414446\pi\)
0.265550 + 0.964097i \(0.414446\pi\)
\(312\) 0 0
\(313\) −8.03742e8 −1.48153 −0.740767 0.671762i \(-0.765538\pi\)
−0.740767 + 0.671762i \(0.765538\pi\)
\(314\) 3.73653e8 0.681105
\(315\) 0 0
\(316\) 2.77973e8 0.495561
\(317\) −2.22203e8 −0.391780 −0.195890 0.980626i \(-0.562760\pi\)
−0.195890 + 0.980626i \(0.562760\pi\)
\(318\) 0 0
\(319\) 1.78988e8 0.308715
\(320\) 3.50557e8 0.598045
\(321\) 0 0
\(322\) −3.96627e8 −0.662043
\(323\) 1.31184e8 0.216607
\(324\) 0 0
\(325\) −7.72971e7 −0.124903
\(326\) −4.90317e8 −0.783818
\(327\) 0 0
\(328\) −8.13259e8 −1.27254
\(329\) 3.38694e8 0.524351
\(330\) 0 0
\(331\) −2.85912e8 −0.433346 −0.216673 0.976244i \(-0.569521\pi\)
−0.216673 + 0.976244i \(0.569521\pi\)
\(332\) −6.98633e8 −1.04777
\(333\) 0 0
\(334\) 5.87348e8 0.862547
\(335\) −1.95025e8 −0.283421
\(336\) 0 0
\(337\) −4.83392e8 −0.688010 −0.344005 0.938968i \(-0.611784\pi\)
−0.344005 + 0.938968i \(0.611784\pi\)
\(338\) −4.26261e8 −0.600436
\(339\) 0 0
\(340\) 3.63783e8 0.501956
\(341\) 1.68440e8 0.230041
\(342\) 0 0
\(343\) −8.01474e8 −1.07241
\(344\) −8.70697e8 −1.15322
\(345\) 0 0
\(346\) −1.70377e8 −0.221129
\(347\) −6.40094e8 −0.822414 −0.411207 0.911542i \(-0.634893\pi\)
−0.411207 + 0.911542i \(0.634893\pi\)
\(348\) 0 0
\(349\) 4.24832e7 0.0534968 0.0267484 0.999642i \(-0.491485\pi\)
0.0267484 + 0.999642i \(0.491485\pi\)
\(350\) 3.25246e7 0.0405483
\(351\) 0 0
\(352\) 5.54266e8 0.677359
\(353\) −1.12511e9 −1.36139 −0.680694 0.732568i \(-0.738322\pi\)
−0.680694 + 0.732568i \(0.738322\pi\)
\(354\) 0 0
\(355\) −1.14971e9 −1.36391
\(356\) −4.59388e8 −0.539640
\(357\) 0 0
\(358\) −7.90408e7 −0.0910460
\(359\) −1.35745e7 −0.0154843 −0.00774215 0.999970i \(-0.502464\pi\)
−0.00774215 + 0.999970i \(0.502464\pi\)
\(360\) 0 0
\(361\) −8.36568e8 −0.935893
\(362\) −7.84090e7 −0.0868733
\(363\) 0 0
\(364\) −5.82061e8 −0.632578
\(365\) −2.14618e8 −0.231015
\(366\) 0 0
\(367\) 8.41482e8 0.888615 0.444308 0.895874i \(-0.353450\pi\)
0.444308 + 0.895874i \(0.353450\pi\)
\(368\) −1.01972e7 −0.0106663
\(369\) 0 0
\(370\) 7.03767e8 0.722309
\(371\) −6.33216e7 −0.0643789
\(372\) 0 0
\(373\) −1.40652e9 −1.40334 −0.701672 0.712500i \(-0.747563\pi\)
−0.701672 + 0.712500i \(0.747563\pi\)
\(374\) −3.64723e8 −0.360506
\(375\) 0 0
\(376\) 7.38386e8 0.716351
\(377\) −6.63554e8 −0.637795
\(378\) 0 0
\(379\) −7.42481e8 −0.700564 −0.350282 0.936644i \(-0.613914\pi\)
−0.350282 + 0.936644i \(0.613914\pi\)
\(380\) 1.58906e8 0.148558
\(381\) 0 0
\(382\) −2.52400e8 −0.231669
\(383\) 7.94805e8 0.722878 0.361439 0.932396i \(-0.382286\pi\)
0.361439 + 0.932396i \(0.382286\pi\)
\(384\) 0 0
\(385\) −5.32296e8 −0.475379
\(386\) −1.95504e8 −0.173021
\(387\) 0 0
\(388\) −1.15773e9 −1.00623
\(389\) 8.55397e8 0.736790 0.368395 0.929669i \(-0.379907\pi\)
0.368395 + 0.929669i \(0.379907\pi\)
\(390\) 0 0
\(391\) −1.47021e9 −1.24383
\(392\) −5.51987e8 −0.462837
\(393\) 0 0
\(394\) 6.30475e8 0.519315
\(395\) −9.42405e8 −0.769392
\(396\) 0 0
\(397\) 3.23507e8 0.259487 0.129744 0.991548i \(-0.458585\pi\)
0.129744 + 0.991548i \(0.458585\pi\)
\(398\) −1.38518e8 −0.110133
\(399\) 0 0
\(400\) 836203. 0.000653284 0
\(401\) 2.15538e8 0.166924 0.0834621 0.996511i \(-0.473402\pi\)
0.0834621 + 0.996511i \(0.473402\pi\)
\(402\) 0 0
\(403\) −6.24449e8 −0.475258
\(404\) −3.82411e8 −0.288533
\(405\) 0 0
\(406\) 2.79206e8 0.207054
\(407\) 1.12591e9 0.827795
\(408\) 0 0
\(409\) 2.07548e9 1.49998 0.749992 0.661447i \(-0.230057\pi\)
0.749992 + 0.661447i \(0.230057\pi\)
\(410\) 1.04968e9 0.752165
\(411\) 0 0
\(412\) 3.64716e8 0.256930
\(413\) −1.36733e8 −0.0955096
\(414\) 0 0
\(415\) 2.36856e9 1.62673
\(416\) −2.05480e9 −1.39940
\(417\) 0 0
\(418\) −1.59317e8 −0.106695
\(419\) 2.46186e9 1.63499 0.817495 0.575936i \(-0.195362\pi\)
0.817495 + 0.575936i \(0.195362\pi\)
\(420\) 0 0
\(421\) 2.15291e9 1.40617 0.703086 0.711105i \(-0.251805\pi\)
0.703086 + 0.711105i \(0.251805\pi\)
\(422\) −1.12089e9 −0.726055
\(423\) 0 0
\(424\) −1.38047e8 −0.0879523
\(425\) 1.20562e8 0.0761814
\(426\) 0 0
\(427\) 7.22497e8 0.449095
\(428\) −1.22707e9 −0.756510
\(429\) 0 0
\(430\) 1.12382e9 0.681640
\(431\) 2.75746e9 1.65897 0.829486 0.558528i \(-0.188634\pi\)
0.829486 + 0.558528i \(0.188634\pi\)
\(432\) 0 0
\(433\) 1.22443e9 0.724814 0.362407 0.932020i \(-0.381955\pi\)
0.362407 + 0.932020i \(0.381955\pi\)
\(434\) 2.62751e8 0.154288
\(435\) 0 0
\(436\) 4.36668e7 0.0252318
\(437\) −6.42212e8 −0.368123
\(438\) 0 0
\(439\) −1.80512e9 −1.01831 −0.509155 0.860675i \(-0.670042\pi\)
−0.509155 + 0.860675i \(0.670042\pi\)
\(440\) −1.16046e9 −0.649448
\(441\) 0 0
\(442\) 1.35212e9 0.744795
\(443\) −2.60973e9 −1.42621 −0.713103 0.701059i \(-0.752711\pi\)
−0.713103 + 0.701059i \(0.752711\pi\)
\(444\) 0 0
\(445\) 1.55745e9 0.837828
\(446\) 8.57609e8 0.457738
\(447\) 0 0
\(448\) 8.74848e8 0.459684
\(449\) −2.04391e9 −1.06561 −0.532805 0.846238i \(-0.678862\pi\)
−0.532805 + 0.846238i \(0.678862\pi\)
\(450\) 0 0
\(451\) 1.67931e9 0.862012
\(452\) −6.39699e7 −0.0325830
\(453\) 0 0
\(454\) −7.42685e7 −0.0372485
\(455\) 1.97335e9 0.982119
\(456\) 0 0
\(457\) −1.13009e9 −0.553866 −0.276933 0.960889i \(-0.589318\pi\)
−0.276933 + 0.960889i \(0.589318\pi\)
\(458\) −3.82638e8 −0.186105
\(459\) 0 0
\(460\) −1.78089e9 −0.853072
\(461\) −1.04379e9 −0.496202 −0.248101 0.968734i \(-0.579806\pi\)
−0.248101 + 0.968734i \(0.579806\pi\)
\(462\) 0 0
\(463\) 3.87197e8 0.181300 0.0906501 0.995883i \(-0.471106\pi\)
0.0906501 + 0.995883i \(0.471106\pi\)
\(464\) 7.17835e6 0.00333589
\(465\) 0 0
\(466\) 7.26055e8 0.332368
\(467\) −3.01337e8 −0.136913 −0.0684563 0.997654i \(-0.521807\pi\)
−0.0684563 + 0.997654i \(0.521807\pi\)
\(468\) 0 0
\(469\) −4.86702e8 −0.217850
\(470\) −9.53040e8 −0.423417
\(471\) 0 0
\(472\) −2.98090e8 −0.130482
\(473\) 1.79791e9 0.781187
\(474\) 0 0
\(475\) 5.26633e7 0.0225465
\(476\) 9.07853e8 0.385826
\(477\) 0 0
\(478\) −6.45888e8 −0.270495
\(479\) 4.36417e9 1.81438 0.907189 0.420723i \(-0.138224\pi\)
0.907189 + 0.420723i \(0.138224\pi\)
\(480\) 0 0
\(481\) −4.17402e9 −1.71020
\(482\) 2.26742e9 0.922288
\(483\) 0 0
\(484\) 8.26604e8 0.331390
\(485\) 3.92502e9 1.56223
\(486\) 0 0
\(487\) 3.93974e9 1.54567 0.772835 0.634608i \(-0.218838\pi\)
0.772835 + 0.634608i \(0.218838\pi\)
\(488\) 1.57511e9 0.613539
\(489\) 0 0
\(490\) 7.12454e8 0.273571
\(491\) 1.95275e9 0.744495 0.372248 0.928133i \(-0.378587\pi\)
0.372248 + 0.928133i \(0.378587\pi\)
\(492\) 0 0
\(493\) 1.03496e9 0.389008
\(494\) 5.90626e8 0.220429
\(495\) 0 0
\(496\) 6.75531e6 0.00248576
\(497\) −2.86920e9 −1.04837
\(498\) 0 0
\(499\) 1.78306e9 0.642414 0.321207 0.947009i \(-0.395912\pi\)
0.321207 + 0.947009i \(0.395912\pi\)
\(500\) 1.78602e9 0.638987
\(501\) 0 0
\(502\) 1.95439e9 0.689522
\(503\) −1.58061e8 −0.0553781 −0.0276890 0.999617i \(-0.508815\pi\)
−0.0276890 + 0.999617i \(0.508815\pi\)
\(504\) 0 0
\(505\) 1.29648e9 0.447967
\(506\) 1.78550e9 0.612679
\(507\) 0 0
\(508\) 3.14019e9 1.06276
\(509\) −3.07905e9 −1.03492 −0.517458 0.855709i \(-0.673122\pi\)
−0.517458 + 0.855709i \(0.673122\pi\)
\(510\) 0 0
\(511\) −5.35599e8 −0.177569
\(512\) 4.45593e7 0.0146721
\(513\) 0 0
\(514\) −1.10221e9 −0.358009
\(515\) −1.23649e9 −0.398901
\(516\) 0 0
\(517\) −1.52470e9 −0.485253
\(518\) 1.75632e9 0.555199
\(519\) 0 0
\(520\) 4.30209e9 1.34174
\(521\) −1.45110e9 −0.449538 −0.224769 0.974412i \(-0.572163\pi\)
−0.224769 + 0.974412i \(0.572163\pi\)
\(522\) 0 0
\(523\) −6.11321e8 −0.186859 −0.0934293 0.995626i \(-0.529783\pi\)
−0.0934293 + 0.995626i \(0.529783\pi\)
\(524\) 3.41159e9 1.03585
\(525\) 0 0
\(526\) 1.70079e9 0.509566
\(527\) 9.73966e8 0.289872
\(528\) 0 0
\(529\) 3.79259e9 1.11389
\(530\) 1.78179e8 0.0519864
\(531\) 0 0
\(532\) 3.96564e8 0.114189
\(533\) −6.22561e9 −1.78089
\(534\) 0 0
\(535\) 4.16010e9 1.17453
\(536\) −1.06106e9 −0.297620
\(537\) 0 0
\(538\) 4.00454e8 0.110870
\(539\) 1.13981e9 0.313524
\(540\) 0 0
\(541\) 4.85916e9 1.31938 0.659691 0.751537i \(-0.270687\pi\)
0.659691 + 0.751537i \(0.270687\pi\)
\(542\) −1.80637e9 −0.487316
\(543\) 0 0
\(544\) 3.20492e9 0.853533
\(545\) −1.48043e8 −0.0391741
\(546\) 0 0
\(547\) −4.87899e9 −1.27460 −0.637300 0.770616i \(-0.719949\pi\)
−0.637300 + 0.770616i \(0.719949\pi\)
\(548\) −2.60892e9 −0.677217
\(549\) 0 0
\(550\) −1.46416e8 −0.0375249
\(551\) 4.52086e8 0.115130
\(552\) 0 0
\(553\) −2.35186e9 −0.591389
\(554\) 4.37186e9 1.09240
\(555\) 0 0
\(556\) 3.75642e9 0.926856
\(557\) −3.74696e9 −0.918725 −0.459362 0.888249i \(-0.651922\pi\)
−0.459362 + 0.888249i \(0.651922\pi\)
\(558\) 0 0
\(559\) −6.66531e9 −1.61391
\(560\) −2.13478e7 −0.00513682
\(561\) 0 0
\(562\) 2.10996e9 0.501414
\(563\) −1.61119e9 −0.380512 −0.190256 0.981735i \(-0.560932\pi\)
−0.190256 + 0.981735i \(0.560932\pi\)
\(564\) 0 0
\(565\) 2.16876e8 0.0505873
\(566\) 3.07394e9 0.712588
\(567\) 0 0
\(568\) −6.25513e9 −1.43224
\(569\) −1.15056e9 −0.261828 −0.130914 0.991394i \(-0.541791\pi\)
−0.130914 + 0.991394i \(0.541791\pi\)
\(570\) 0 0
\(571\) −4.25054e9 −0.955472 −0.477736 0.878503i \(-0.658543\pi\)
−0.477736 + 0.878503i \(0.658543\pi\)
\(572\) 2.62027e9 0.585410
\(573\) 0 0
\(574\) 2.61957e9 0.578148
\(575\) −5.90209e8 −0.129470
\(576\) 0 0
\(577\) 2.31080e9 0.500780 0.250390 0.968145i \(-0.419441\pi\)
0.250390 + 0.968145i \(0.419441\pi\)
\(578\) 7.72580e8 0.166416
\(579\) 0 0
\(580\) 1.25366e9 0.266798
\(581\) 5.91096e9 1.25038
\(582\) 0 0
\(583\) 2.85056e8 0.0595785
\(584\) −1.16766e9 −0.242589
\(585\) 0 0
\(586\) −2.76948e9 −0.568534
\(587\) 4.19562e9 0.856175 0.428087 0.903737i \(-0.359188\pi\)
0.428087 + 0.903737i \(0.359188\pi\)
\(588\) 0 0
\(589\) 4.25443e8 0.0857902
\(590\) 3.84748e8 0.0771248
\(591\) 0 0
\(592\) 4.51547e7 0.00894493
\(593\) 7.48887e9 1.47477 0.737386 0.675472i \(-0.236060\pi\)
0.737386 + 0.675472i \(0.236060\pi\)
\(594\) 0 0
\(595\) −3.07788e9 −0.599020
\(596\) −6.22611e9 −1.20463
\(597\) 0 0
\(598\) −6.61928e9 −1.26578
\(599\) −2.12227e9 −0.403467 −0.201733 0.979440i \(-0.564657\pi\)
−0.201733 + 0.979440i \(0.564657\pi\)
\(600\) 0 0
\(601\) −6.18903e8 −0.116295 −0.0581477 0.998308i \(-0.518519\pi\)
−0.0581477 + 0.998308i \(0.518519\pi\)
\(602\) 2.80459e9 0.523939
\(603\) 0 0
\(604\) 4.13183e9 0.762979
\(605\) −2.80242e9 −0.514504
\(606\) 0 0
\(607\) −4.99735e8 −0.0906942 −0.0453471 0.998971i \(-0.514439\pi\)
−0.0453471 + 0.998971i \(0.514439\pi\)
\(608\) 1.39996e9 0.252611
\(609\) 0 0
\(610\) −2.03301e9 −0.362648
\(611\) 5.65245e9 1.00252
\(612\) 0 0
\(613\) 7.99751e9 1.40231 0.701154 0.713010i \(-0.252669\pi\)
0.701154 + 0.713010i \(0.252669\pi\)
\(614\) 1.93371e9 0.337133
\(615\) 0 0
\(616\) −2.89603e9 −0.499195
\(617\) −1.58063e9 −0.270914 −0.135457 0.990783i \(-0.543250\pi\)
−0.135457 + 0.990783i \(0.543250\pi\)
\(618\) 0 0
\(619\) 6.43846e9 1.09110 0.545550 0.838078i \(-0.316321\pi\)
0.545550 + 0.838078i \(0.316321\pi\)
\(620\) 1.17978e9 0.198806
\(621\) 0 0
\(622\) −1.97840e9 −0.329646
\(623\) 3.88677e9 0.643992
\(624\) 0 0
\(625\) −5.51161e9 −0.903022
\(626\) 5.64409e9 0.919568
\(627\) 0 0
\(628\) 4.18695e9 0.674589
\(629\) 6.51030e9 1.04310
\(630\) 0 0
\(631\) −1.07401e10 −1.70179 −0.850896 0.525334i \(-0.823940\pi\)
−0.850896 + 0.525334i \(0.823940\pi\)
\(632\) −5.12728e9 −0.807936
\(633\) 0 0
\(634\) 1.56037e9 0.243173
\(635\) −1.06461e10 −1.65000
\(636\) 0 0
\(637\) −4.22554e9 −0.647731
\(638\) −1.25690e9 −0.191615
\(639\) 0 0
\(640\) 3.85335e9 0.581043
\(641\) −4.76542e9 −0.714658 −0.357329 0.933978i \(-0.616313\pi\)
−0.357329 + 0.933978i \(0.616313\pi\)
\(642\) 0 0
\(643\) −5.32986e9 −0.790637 −0.395319 0.918544i \(-0.629366\pi\)
−0.395319 + 0.918544i \(0.629366\pi\)
\(644\) −4.44439e9 −0.655709
\(645\) 0 0
\(646\) −9.21211e8 −0.134445
\(647\) −3.64298e9 −0.528801 −0.264400 0.964413i \(-0.585174\pi\)
−0.264400 + 0.964413i \(0.585174\pi\)
\(648\) 0 0
\(649\) 6.15531e8 0.0883881
\(650\) 5.42801e8 0.0775253
\(651\) 0 0
\(652\) −5.49423e9 −0.776319
\(653\) −3.76641e9 −0.529337 −0.264668 0.964340i \(-0.585262\pi\)
−0.264668 + 0.964340i \(0.585262\pi\)
\(654\) 0 0
\(655\) −1.15662e10 −1.60823
\(656\) 6.73490e7 0.00931467
\(657\) 0 0
\(658\) −2.37840e9 −0.325458
\(659\) 4.55405e9 0.619867 0.309934 0.950758i \(-0.399693\pi\)
0.309934 + 0.950758i \(0.399693\pi\)
\(660\) 0 0
\(661\) −1.46628e9 −0.197475 −0.0987377 0.995113i \(-0.531480\pi\)
−0.0987377 + 0.995113i \(0.531480\pi\)
\(662\) 2.00775e9 0.268972
\(663\) 0 0
\(664\) 1.28865e10 1.70823
\(665\) −1.34446e9 −0.177285
\(666\) 0 0
\(667\) −5.06663e9 −0.661118
\(668\) 6.58151e9 0.854295
\(669\) 0 0
\(670\) 1.36952e9 0.175916
\(671\) −3.25247e9 −0.415609
\(672\) 0 0
\(673\) 1.29610e10 1.63903 0.819514 0.573059i \(-0.194244\pi\)
0.819514 + 0.573059i \(0.194244\pi\)
\(674\) 3.39451e9 0.427039
\(675\) 0 0
\(676\) −4.77645e9 −0.594692
\(677\) −2.05035e9 −0.253962 −0.126981 0.991905i \(-0.540529\pi\)
−0.126981 + 0.991905i \(0.540529\pi\)
\(678\) 0 0
\(679\) 9.79525e9 1.20080
\(680\) −6.71007e9 −0.818362
\(681\) 0 0
\(682\) −1.18283e9 −0.142783
\(683\) −1.07567e9 −0.129184 −0.0645918 0.997912i \(-0.520575\pi\)
−0.0645918 + 0.997912i \(0.520575\pi\)
\(684\) 0 0
\(685\) 8.84495e9 1.05143
\(686\) 5.62817e9 0.665629
\(687\) 0 0
\(688\) 7.21056e7 0.00844130
\(689\) −1.05677e9 −0.123087
\(690\) 0 0
\(691\) 4.59832e9 0.530183 0.265092 0.964223i \(-0.414598\pi\)
0.265092 + 0.964223i \(0.414598\pi\)
\(692\) −1.90916e9 −0.219013
\(693\) 0 0
\(694\) 4.49491e9 0.510461
\(695\) −1.27353e10 −1.43901
\(696\) 0 0
\(697\) 9.71022e9 1.08621
\(698\) −2.98328e8 −0.0332048
\(699\) 0 0
\(700\) 3.64453e8 0.0401604
\(701\) −9.38304e9 −1.02880 −0.514400 0.857551i \(-0.671985\pi\)
−0.514400 + 0.857551i \(0.671985\pi\)
\(702\) 0 0
\(703\) 2.84380e9 0.308713
\(704\) −3.93832e9 −0.425409
\(705\) 0 0
\(706\) 7.90079e9 0.844995
\(707\) 3.23549e9 0.344327
\(708\) 0 0
\(709\) −1.12461e10 −1.18506 −0.592528 0.805550i \(-0.701870\pi\)
−0.592528 + 0.805550i \(0.701870\pi\)
\(710\) 8.07354e9 0.846564
\(711\) 0 0
\(712\) 8.47353e9 0.879800
\(713\) −4.76804e9 −0.492636
\(714\) 0 0
\(715\) −8.88345e9 −0.908889
\(716\) −8.85689e8 −0.0901750
\(717\) 0 0
\(718\) 9.53234e7 0.00961090
\(719\) 6.98063e9 0.700396 0.350198 0.936676i \(-0.386114\pi\)
0.350198 + 0.936676i \(0.386114\pi\)
\(720\) 0 0
\(721\) −3.08578e9 −0.306613
\(722\) 5.87461e9 0.580896
\(723\) 0 0
\(724\) −8.78610e8 −0.0860421
\(725\) 4.15478e8 0.0404917
\(726\) 0 0
\(727\) 7.76937e9 0.749921 0.374960 0.927041i \(-0.377656\pi\)
0.374960 + 0.927041i \(0.377656\pi\)
\(728\) 1.07363e10 1.03132
\(729\) 0 0
\(730\) 1.50710e9 0.143388
\(731\) 1.03960e10 0.984366
\(732\) 0 0
\(733\) −1.54005e10 −1.44434 −0.722172 0.691713i \(-0.756856\pi\)
−0.722172 + 0.691713i \(0.756856\pi\)
\(734\) −5.90911e9 −0.551551
\(735\) 0 0
\(736\) −1.56896e10 −1.45058
\(737\) 2.19099e9 0.201607
\(738\) 0 0
\(739\) 8.56257e9 0.780456 0.390228 0.920718i \(-0.372396\pi\)
0.390228 + 0.920718i \(0.372396\pi\)
\(740\) 7.88604e9 0.715398
\(741\) 0 0
\(742\) 4.44661e8 0.0399591
\(743\) 3.55238e9 0.317730 0.158865 0.987300i \(-0.449217\pi\)
0.158865 + 0.987300i \(0.449217\pi\)
\(744\) 0 0
\(745\) 2.11083e10 1.87027
\(746\) 9.87694e9 0.871037
\(747\) 0 0
\(748\) −4.08689e9 −0.357057
\(749\) 1.03819e10 0.902798
\(750\) 0 0
\(751\) 1.09272e10 0.941391 0.470696 0.882296i \(-0.344003\pi\)
0.470696 + 0.882296i \(0.344003\pi\)
\(752\) −6.11484e7 −0.00524352
\(753\) 0 0
\(754\) 4.65965e9 0.395871
\(755\) −1.40080e10 −1.18458
\(756\) 0 0
\(757\) 6.79624e9 0.569421 0.284710 0.958614i \(-0.408103\pi\)
0.284710 + 0.958614i \(0.408103\pi\)
\(758\) 5.21390e9 0.434831
\(759\) 0 0
\(760\) −2.93106e9 −0.242202
\(761\) −1.52139e10 −1.25139 −0.625696 0.780067i \(-0.715185\pi\)
−0.625696 + 0.780067i \(0.715185\pi\)
\(762\) 0 0
\(763\) −3.69454e8 −0.0301109
\(764\) −2.82826e9 −0.229452
\(765\) 0 0
\(766\) −5.58133e9 −0.448680
\(767\) −2.28192e9 −0.182607
\(768\) 0 0
\(769\) −1.81575e10 −1.43984 −0.719921 0.694056i \(-0.755822\pi\)
−0.719921 + 0.694056i \(0.755822\pi\)
\(770\) 3.73792e9 0.295062
\(771\) 0 0
\(772\) −2.19071e9 −0.171366
\(773\) −1.26937e10 −0.988459 −0.494230 0.869331i \(-0.664550\pi\)
−0.494230 + 0.869331i \(0.664550\pi\)
\(774\) 0 0
\(775\) 3.90993e8 0.0301726
\(776\) 2.13546e10 1.64050
\(777\) 0 0
\(778\) −6.00682e9 −0.457316
\(779\) 4.24157e9 0.321474
\(780\) 0 0
\(781\) 1.29163e10 0.970196
\(782\) 1.03242e10 0.772030
\(783\) 0 0
\(784\) 4.57121e7 0.00338786
\(785\) −1.41949e10 −1.04734
\(786\) 0 0
\(787\) −2.24532e10 −1.64198 −0.820988 0.570946i \(-0.806577\pi\)
−0.820988 + 0.570946i \(0.806577\pi\)
\(788\) 7.06477e9 0.514347
\(789\) 0 0
\(790\) 6.61782e9 0.477551
\(791\) 5.41234e8 0.0388837
\(792\) 0 0
\(793\) 1.20577e10 0.858635
\(794\) −2.27175e9 −0.161060
\(795\) 0 0
\(796\) −1.55216e9 −0.109079
\(797\) −4.23290e9 −0.296165 −0.148083 0.988975i \(-0.547310\pi\)
−0.148083 + 0.988975i \(0.547310\pi\)
\(798\) 0 0
\(799\) −8.81624e9 −0.611462
\(800\) 1.28660e9 0.0888438
\(801\) 0 0
\(802\) −1.51357e9 −0.103608
\(803\) 2.41111e9 0.164328
\(804\) 0 0
\(805\) 1.50677e10 1.01803
\(806\) 4.38504e9 0.294986
\(807\) 0 0
\(808\) 7.05367e9 0.470409
\(809\) 5.59705e9 0.371654 0.185827 0.982582i \(-0.440504\pi\)
0.185827 + 0.982582i \(0.440504\pi\)
\(810\) 0 0
\(811\) 2.20508e10 1.45162 0.725808 0.687897i \(-0.241466\pi\)
0.725808 + 0.687897i \(0.241466\pi\)
\(812\) 3.12863e9 0.205073
\(813\) 0 0
\(814\) −7.90643e9 −0.513801
\(815\) 1.86270e10 1.20529
\(816\) 0 0
\(817\) 4.54114e9 0.291332
\(818\) −1.45746e10 −0.931020
\(819\) 0 0
\(820\) 1.17622e10 0.744969
\(821\) −1.92550e10 −1.21435 −0.607173 0.794569i \(-0.707697\pi\)
−0.607173 + 0.794569i \(0.707697\pi\)
\(822\) 0 0
\(823\) −2.63648e10 −1.64864 −0.824318 0.566127i \(-0.808441\pi\)
−0.824318 + 0.566127i \(0.808441\pi\)
\(824\) −6.72729e9 −0.418885
\(825\) 0 0
\(826\) 9.60173e8 0.0592815
\(827\) −2.32586e10 −1.42993 −0.714965 0.699160i \(-0.753557\pi\)
−0.714965 + 0.699160i \(0.753557\pi\)
\(828\) 0 0
\(829\) 1.16870e10 0.712464 0.356232 0.934398i \(-0.384061\pi\)
0.356232 + 0.934398i \(0.384061\pi\)
\(830\) −1.66326e10 −1.00969
\(831\) 0 0
\(832\) 1.46003e10 0.878881
\(833\) 6.59067e9 0.395068
\(834\) 0 0
\(835\) −2.23132e10 −1.32635
\(836\) −1.78522e9 −0.105674
\(837\) 0 0
\(838\) −1.72879e10 −1.01482
\(839\) −3.29655e10 −1.92705 −0.963524 0.267623i \(-0.913762\pi\)
−0.963524 + 0.267623i \(0.913762\pi\)
\(840\) 0 0
\(841\) −1.36832e10 −0.793236
\(842\) −1.51183e10 −0.872791
\(843\) 0 0
\(844\) −1.25601e10 −0.719109
\(845\) 1.61935e10 0.923299
\(846\) 0 0
\(847\) −6.99369e9 −0.395471
\(848\) 1.14322e7 0.000643790 0
\(849\) 0 0
\(850\) −8.46617e8 −0.0472847
\(851\) −3.18711e10 −1.77274
\(852\) 0 0
\(853\) −6.75557e9 −0.372684 −0.186342 0.982485i \(-0.559663\pi\)
−0.186342 + 0.982485i \(0.559663\pi\)
\(854\) −5.07356e9 −0.278747
\(855\) 0 0
\(856\) 2.26336e10 1.23337
\(857\) −3.29149e10 −1.78632 −0.893162 0.449735i \(-0.851518\pi\)
−0.893162 + 0.449735i \(0.851518\pi\)
\(858\) 0 0
\(859\) 1.99567e10 1.07427 0.537135 0.843496i \(-0.319506\pi\)
0.537135 + 0.843496i \(0.319506\pi\)
\(860\) 1.25929e10 0.675119
\(861\) 0 0
\(862\) −1.93636e10 −1.02970
\(863\) −1.08133e10 −0.572689 −0.286344 0.958127i \(-0.592440\pi\)
−0.286344 + 0.958127i \(0.592440\pi\)
\(864\) 0 0
\(865\) 6.47257e9 0.340033
\(866\) −8.59828e9 −0.449882
\(867\) 0 0
\(868\) 2.94425e9 0.152811
\(869\) 1.05874e10 0.547293
\(870\) 0 0
\(871\) −8.12255e9 −0.416513
\(872\) −8.05445e8 −0.0411366
\(873\) 0 0
\(874\) 4.50978e9 0.228489
\(875\) −1.51111e10 −0.762549
\(876\) 0 0
\(877\) 3.04464e10 1.52418 0.762091 0.647470i \(-0.224173\pi\)
0.762091 + 0.647470i \(0.224173\pi\)
\(878\) 1.26760e10 0.632051
\(879\) 0 0
\(880\) 9.61016e7 0.00475380
\(881\) 1.83036e10 0.901821 0.450910 0.892569i \(-0.351099\pi\)
0.450910 + 0.892569i \(0.351099\pi\)
\(882\) 0 0
\(883\) −2.58285e10 −1.26251 −0.631257 0.775574i \(-0.717461\pi\)
−0.631257 + 0.775574i \(0.717461\pi\)
\(884\) 1.51511e10 0.737669
\(885\) 0 0
\(886\) 1.83262e10 0.885227
\(887\) 1.56417e10 0.752578 0.376289 0.926502i \(-0.377200\pi\)
0.376289 + 0.926502i \(0.377200\pi\)
\(888\) 0 0
\(889\) −2.65684e10 −1.26826
\(890\) −1.09368e10 −0.520028
\(891\) 0 0
\(892\) 9.60991e9 0.453359
\(893\) −3.85107e9 −0.180968
\(894\) 0 0
\(895\) 3.00273e9 0.140003
\(896\) 9.61638e9 0.446615
\(897\) 0 0
\(898\) 1.43528e10 0.661410
\(899\) 3.35647e9 0.154072
\(900\) 0 0
\(901\) 1.64827e9 0.0750743
\(902\) −1.17926e10 −0.535039
\(903\) 0 0
\(904\) 1.17994e9 0.0531216
\(905\) 2.97873e9 0.133586
\(906\) 0 0
\(907\) 3.32082e10 1.47781 0.738906 0.673808i \(-0.235343\pi\)
0.738906 + 0.673808i \(0.235343\pi\)
\(908\) −8.32213e8 −0.0368922
\(909\) 0 0
\(910\) −1.38574e10 −0.609588
\(911\) 6.12880e9 0.268572 0.134286 0.990943i \(-0.457126\pi\)
0.134286 + 0.990943i \(0.457126\pi\)
\(912\) 0 0
\(913\) −2.66094e10 −1.15715
\(914\) 7.93576e9 0.343777
\(915\) 0 0
\(916\) −4.28764e9 −0.184325
\(917\) −2.88646e10 −1.23616
\(918\) 0 0
\(919\) 1.60841e10 0.683585 0.341793 0.939775i \(-0.388966\pi\)
0.341793 + 0.939775i \(0.388966\pi\)
\(920\) 3.28491e10 1.39080
\(921\) 0 0
\(922\) 7.32974e9 0.307986
\(923\) −4.78839e10 −2.00439
\(924\) 0 0
\(925\) 2.61353e9 0.108575
\(926\) −2.71900e9 −0.112531
\(927\) 0 0
\(928\) 1.10447e10 0.453667
\(929\) 1.02473e10 0.419327 0.209664 0.977774i \(-0.432763\pi\)
0.209664 + 0.977774i \(0.432763\pi\)
\(930\) 0 0
\(931\) 2.87890e9 0.116924
\(932\) 8.13579e9 0.329188
\(933\) 0 0
\(934\) 2.11607e9 0.0849798
\(935\) 1.38557e10 0.554355
\(936\) 0 0
\(937\) −1.95793e10 −0.777516 −0.388758 0.921340i \(-0.627096\pi\)
−0.388758 + 0.921340i \(0.627096\pi\)
\(938\) 3.41775e9 0.135217
\(939\) 0 0
\(940\) −1.06793e10 −0.419367
\(941\) −7.05944e8 −0.0276189 −0.0138095 0.999905i \(-0.504396\pi\)
−0.0138095 + 0.999905i \(0.504396\pi\)
\(942\) 0 0
\(943\) −4.75363e10 −1.84601
\(944\) 2.46860e7 0.000955098 0
\(945\) 0 0
\(946\) −1.26254e10 −0.484872
\(947\) −1.93347e10 −0.739798 −0.369899 0.929072i \(-0.620608\pi\)
−0.369899 + 0.929072i \(0.620608\pi\)
\(948\) 0 0
\(949\) −8.93858e9 −0.339498
\(950\) −3.69815e8 −0.0139943
\(951\) 0 0
\(952\) −1.67456e10 −0.629030
\(953\) 2.96357e10 1.10915 0.554574 0.832134i \(-0.312881\pi\)
0.554574 + 0.832134i \(0.312881\pi\)
\(954\) 0 0
\(955\) 9.58859e9 0.356240
\(956\) −7.23747e9 −0.267907
\(957\) 0 0
\(958\) −3.06464e10 −1.12616
\(959\) 2.20734e10 0.808172
\(960\) 0 0
\(961\) −2.43540e10 −0.885192
\(962\) 2.93111e10 1.06150
\(963\) 0 0
\(964\) 2.54075e10 0.913465
\(965\) 7.42712e9 0.266057
\(966\) 0 0
\(967\) −2.65671e10 −0.944824 −0.472412 0.881378i \(-0.656617\pi\)
−0.472412 + 0.881378i \(0.656617\pi\)
\(968\) −1.52469e10 −0.540280
\(969\) 0 0
\(970\) −2.75625e10 −0.969657
\(971\) 1.88208e10 0.659736 0.329868 0.944027i \(-0.392996\pi\)
0.329868 + 0.944027i \(0.392996\pi\)
\(972\) 0 0
\(973\) −3.17821e10 −1.10608
\(974\) −2.76659e10 −0.959376
\(975\) 0 0
\(976\) −1.30441e8 −0.00449096
\(977\) −1.00691e10 −0.345430 −0.172715 0.984972i \(-0.555254\pi\)
−0.172715 + 0.984972i \(0.555254\pi\)
\(978\) 0 0
\(979\) −1.74971e10 −0.595973
\(980\) 7.98338e9 0.270954
\(981\) 0 0
\(982\) −1.37127e10 −0.462098
\(983\) 4.43932e10 1.49066 0.745332 0.666694i \(-0.232291\pi\)
0.745332 + 0.666694i \(0.232291\pi\)
\(984\) 0 0
\(985\) −2.39515e10 −0.798558
\(986\) −7.26775e9 −0.241452
\(987\) 0 0
\(988\) 6.61824e9 0.218320
\(989\) −5.08936e10 −1.67292
\(990\) 0 0
\(991\) 1.29610e10 0.423040 0.211520 0.977374i \(-0.432159\pi\)
0.211520 + 0.977374i \(0.432159\pi\)
\(992\) 1.03938e10 0.338053
\(993\) 0 0
\(994\) 2.01483e10 0.650707
\(995\) 5.26225e9 0.169352
\(996\) 0 0
\(997\) −5.75699e10 −1.83977 −0.919883 0.392194i \(-0.871716\pi\)
−0.919883 + 0.392194i \(0.871716\pi\)
\(998\) −1.25211e10 −0.398737
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 531.8.a.b.1.6 16
3.2 odd 2 177.8.a.a.1.11 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.a.1.11 16 3.2 odd 2
531.8.a.b.1.6 16 1.1 even 1 trivial